We report on a blinded search for dark matter with single- and few-electron signals in the first science run of XENONnT relying on a novel detector response framework that is physics model dependent. We derive 90% confidence upper limits for dark matter-electron interactions. Heavy and light mediator cases are considered for the standard halo model and dark matter up-scattered in the Sun. We set stringent new limits on dark matter-electron scattering via a heavy mediator with a mass within 10-20 MeV/c^{2} and electron absorption of axionlike particles and dark photons for m_{χ} below 0.03 keV/c^{2}.